While all sorts of things get used in experiments, screws really intended to tap into plastic have larger more aggressive threads than those intended to go into a pre-tapped hole in a harder material like metal.
Generally speaking, you want the screw's
minor diameter (which is to say that of the body between the threads) to fit through the hole, with only the threads cutting into the walls. If the body itself has to be forced, breaking the plastic horn is more common.
It is also quite commonplance to drill out servo horn holes to a slightly larger size when needed - typically you would do this by rotating the drill bit by hand in a pin vise, not with a an electric drill. Of course enlarging the hole weakens the servo horn.
Another option is to buy a miniature machine screw and matching nylon insert nut to install on the reverse side. You may find it easier to install if you remove the servo horn from the servo.
Traditionally hobby shops had lots of small generic fasteners; that seems to be less true today with the prevalence of ready-to-go models vs kits assembled with components purchased individually. However sometimes you can buy a full set of replacement screws for an RC car or helicopter, which will give you a variety of sizes to choose from.